That is incorrect. Sprint does have phones that is capible of PTT and are called Hybrid or Powersource phones.
A perfect example would be the Motorola Rennagade or the Sanyo Pro 700. Sprint just does not sell them on sprint.com, you should be able to find some of them at a retail store.
Sprint discontinued sales of QChat phones last Nov 29th or shortly thereafter as far as I know, I doubt you'd be able to find one in any Sprint store. Maybe ebay, craigslist. They are no longer developing any more so the original question of if it's available for the Rumor Touch is still no.
Android phones do have an app available similiar to PTT, It's called TIKL
Sprint phones that have QChat
I have had the pro 700 going on four years now, and just recently been able to use the DC feature. Do you think Spint is going to come out with new powersource phones now that the service is available?
Future of push-to-talk
. As the Network Vision transformation unfolds, Sprint expects to launch the next-generation of PTT services in 2011 on the CDMA network, offering customers sub-second call setup time along with robust data capabilities.
This is taken for a December 06 2010 news release watch for further informtion